Mathematics – Numerical Analysis
Scientific paper
2012-03-23
Mathematics
Numerical Analysis
This is to replace v1. Some errors (in cpu time) and typos have been corrected. In v2, numerical tests were done using normali
Scientific paper
Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real symmetric tensor and ${\mathcal B}$ be a positive definite tensor of the same size. $\lambda \in R$ is called a ${\mathcal B}_r$-eigenvalue of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ for some $x \in R^n \backslash \{0\}$. In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum ${\mathcal B}_r$--eigenvalues of ${\mathcal A}$. These unconstrained optimization problems can be solved using some powerful optimization algorithms, such as the BFGS method. We provide some numerical results to illustrate the effectiveness of this approach for finding Z-eigenvalues of ${\mathcal A}$.
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